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3/(x-5)+8/x=2

3/(x-5)+8/x=2 equation

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Numerical solution:

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The solution

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  3     8    
----- + - = 2
x - 5   x    
$$\frac{3}{x - 5} + \frac{8}{x} = 2$$
Detail solution
Given the equation:
$$\frac{3}{x - 5} + \frac{8}{x} = 2$$
Multiply the equation sides by the denominators:
x and -5 + x
we get:
$$x \left(\frac{3}{x - 5} + \frac{8}{x}\right) = 2 x$$
$$\frac{11 x - 40}{x - 5} = 2 x$$
$$\frac{11 x - 40}{x - 5} \left(x - 5\right) = 2 x \left(x - 5\right)$$
$$11 x - 40 = 2 x^{2} - 10 x$$
Move right part of the equation to
left part with negative sign.

The equation is transformed from
$$11 x - 40 = 2 x^{2} - 10 x$$
to
$$- 2 x^{2} + 21 x - 40 = 0$$
This equation is of the form
a*x^2 + b*x + c = 0

A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$x_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = -2$$
$$b = 21$$
$$c = -40$$
, then
D = b^2 - 4 * a * c = 

(21)^2 - 4 * (-2) * (-40) = 121

Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)

x2 = (-b - sqrt(D)) / (2*a)

or
$$x_{1} = \frac{5}{2}$$
$$x_{2} = 8$$
The graph
Sum and product of roots [src]
sum
8 + 5/2
$$\frac{5}{2} + 8$$
=
21/2
$$\frac{21}{2}$$
product
8*5
---
 2 
$$\frac{5 \cdot 8}{2}$$
=
20
$$20$$
20
Rapid solution [src]
x1 = 5/2
$$x_{1} = \frac{5}{2}$$
x2 = 8
$$x_{2} = 8$$
x2 = 8
Numerical answer [src]
x1 = 8.0
x2 = 2.5
x2 = 2.5
The graph
3/(x-5)+8/x=2 equation